Problem 124 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the next section. $$ \text { Simplify: } \frac{-\frac{3 \pi}{4}+\frac{\pi}{4}}{2} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(-\frac{1}{4}\pi\).

Step 1: Combine Similar Terms in Numerator

In the numerator, you have \(-\frac{3 \pi}{4} + \frac{\pi}{4}\). Notice both fractions have same denominator and the numerator has the same constant \(\pi\). This allows them to be combined. So, you can simply subtract the two fractions in the numerator. That gives you \(-\frac{3}{4}\pi + \frac{1}{4}\pi = \(-\frac{2}{4}\pi\) which simplifies to \(-\frac{1}{2}\pi\).

Step 2: Divide the Numerator by Denominator

After obtaining \(-\frac{1}{2}\pi\) in the numerator, you then divide this by the denominator, which is 2. So, you have \(\frac{-\frac{1}{2}\pi}{2}\). Dividing a number by 2 is the same as multiplying it by \(\frac{1}{2}\), hence the expression becomes \(-\frac{1}{2}\pi * \frac{1}{2}\).

Step 3: Final Simplification

The multiplication of \(-\frac{1}{2}\pi * \frac{1}{2}\) gives \(-\frac{1}{4}\pi\).

Key Concepts

Numerator and Denominator

When simplifying an algebraic fraction, it is crucial to understand the roles of the numerator and denominator. The numerator refers to the top part of a fraction and the denominator is the bottom part. For the given problem, the fraction is set as \(-\frac{3 \pi}{4} + \frac{\pi}{4}\) over 2. This means the numerator is \(-\frac{3 \pi}{4} + \frac{\pi}{4}\), while the denominator is 2.

Understanding this helps us know where to apply different mathematical operations. Operations such as combining like terms and division are to be performed on the numerator before the entire expression is divided by the denominator.

Combining Like Terms

Combining like terms is an essential skill in algebra, which simplifies expressions by merging terms that are alike. This can include combining terms that share identical variables and matching coefficients. In our problem, the numerator is \(-\frac{3 \pi}{4} + \frac{\pi}{4}\). Notice how both terms have a \(\pi\) variable and they share the same denominator, 4.

Combining them involves adding the coefficients, which are the numbers in front of \(\pi\).

• Multiply the coefficients with their respective denominators.
• Add or subtract to combine them: here it's \(-3 + 1\).
• This leaves us with \(-\frac{2}{4}\pi\), which simplifies to \(-\frac{1}{2}\pi\).

By simplifying, we've reduced the expression and prepared it for further operations.

Fraction Division

After simplifying the numerator, the next step is division by the denominator, 2. This process involves dividing \(-\frac{1}{2}\pi\) by 2. In mathematics, dividing by a number is the same as multiplying by its reciprocal. The reciprocal of 2 is \(\frac{1}{2}\).

This operation is simply a multiplication:

• Take the simplified numerator, \(-\frac{1}{2}\pi\).
• Multiply it by the reciprocal, \(\frac{1}{2}\).
• This calculation \(-\frac{1}{2}\pi * \frac{1}{2}\) becomes \(-\frac{1}{4}\pi\).

By completing this step, you achieve the final simplified form of the initial algebraic fraction. Understanding these steps helps streamline the full process and grasp how algebraic expressions are methodically reduced.